Question

The measure of angle \(\theta\) in standard position is \(4900^{\circ}\) a) Describe \(\theta\) in terms of revolutions. Be specific. b) In which quadrant does \(4900^{\circ}\) terminate? c) What is the measure of the reference angle? d) Give the value of each trigonometric ratio for \(4900^{\circ}\)

Short answer

a) 13.611 revolutions b) Quadrant III c) 40 degrees d) \( \sin(220^{\circ}) = -\sin(40^{\circ}), \cos(220^{\circ}) = -\cos(40^{\circ}), \tan(220^{\circ}) = \tan(40^{\circ}), \csc(220^{\circ}) = -\csc(40^{\circ}), \sec(220^{\circ}) = -\sec(40^{\circ}), \cot(220^{\circ}) = \cot(40^{\circ}) \)

Step-by-step solution

Convert Degrees to Revolutions

Since one full revolution corresponds to 360 degrees, the number of revolutions is calculated by dividing the given angle by 360. Hence, \ \( \text{Number of Revolutions} = \frac{4900}{360} \approx 13.611 \)

Determine the Equivalent Angle in the First Revolution

The equivalent angle in the first revolution can be found by calculating the remainder when 4900 is divided by 360, as each complete 360-degree rotation returns to the start. Thus, \ \(4900 \mod 360 = 220 \) So, the equivalent angle is 220 degrees.

Identify the Quadrant

Recall the boundaries of each quadrant: - Quadrant I: \(0^{\circ}\) to \(90^{\circ}\) - Quadrant II: \(90^{\circ}\) to \(180^{\circ}\) - Quadrant III: \(180^{\circ}\) to \(270^{\circ}\) - Quadrant IV: \(270^{\circ}\) to \(360^{\circ}\) Since 220 degrees falls between 180 and 270 degrees, it terminates in Quadrant III.

Calculate the Reference Angle

The reference angle is the acute angle that the terminal side of the given angle makes with the x-axis. For Quadrant III: \ \( \text{Reference Angle} = 220^{\circ} - 180^{\circ} = 40^{\circ} \)

Determine Trigonometric Ratios

In Quadrant III, the sine and cosine values are both negative. Therefore, let us calculate the trigonometric ratios for \(40^{\circ}\) and then assign proper signs: 1. \( \sin(220^{\circ}) = -\sin(40^{\circ}) \) 2. \( \cos(220^{\circ}) = -\cos(40^{\circ}) \) 3. \( \tan(220^{\circ}) = \tan(40^{\circ}) \) (since both sine and cosine are negative, the tangent ratio remains positive) 4. \( \csc(220^{\circ}) = -\csc(40^{\circ}) \) 5. \( \sec(220^{\circ}) = -\sec(40^{\circ}) \) 6. \( \cot(220^{\circ}) = \cot(40^{\circ}) \)

Key concepts

Angle Measurement

When dealing with angles, the unit we typically use is degrees. A full circle is 360 degrees. Measuring an angle in the standard position means drawing the angle starting from the positive x-axis. If you have an angle like \(4900^{\circ}\), it's a very large angle.

To understand such large angles, we often convert them into smaller, more manageable ones by taking full circles out of them. This process involves dividing the angle by 360 degrees to find the full revolutions. For example, \(4900^{\circ}\) results in about 13.611 revolutions. This tells us that the angle has gone around the circle over 13 times. A smaller equivalent angle within a single circle can be found by taking the remainder, which is obtained by dividing and then taking the modulus \(4900 \mod 360 = 220\). This leftover angle, \(220^{\circ}\), is the same as \(4900^{\circ}\) in terms of its terminal side position.

Revolutions

A revolution is a term used when an angle makes a full circle. One revolution equals 360 degrees.

When converting a large angle into revolutions, we simply divide by 360. For example, converting \(4900^{\circ}\) results in \(4900/360\), which is approximately 13.611 revolutions. This tells us the angle wraps around the circle 13 full times and partially once more.

Understanding revolutions helps visualize large angles. For instance, someone could say 'over 13 complete circles plus a bit more.' The 'bit more' is what's left after accounting for the complete circles, which helps us identify the smaller angle within one circle.

Reference Angle

The reference angle is a simpler angle that helps us understand the trigonometric functions of larger angles. It is always an acute angle (less than 90 degrees). It is the angle between the terminal side of a given angle and the x-axis.

To find the reference angle, we first determine the equivalent angle within a single rotation (0 to 360 degrees). For \(4900^{\circ}\), this is \(220^{\circ}\). We then use the quadrant it falls into to find the reference angle. Since \(220^{\circ}\) is in Quadrant III (180 to 270 degrees), the reference angle is calculated by \(220^{\circ} - 180^{\circ} = 40^{\circ}\).

The reference angle greatly simplifies the calculation of trigonometric functions.

Trigonometric Functions

Trigonometric functions relate the angles of a triangle to the lengths of its sides. For an angle in standard position, depending on which quadrant it terminates in, the signs of these functions will vary.

The main trigonometric functions include sine (\( \text{sin} \)), cosine (\( \text{cos} \)), and tangent (\( \text{tan} \)). Their reciprocal functions are cosecant (\( \text{csc} \)), secant (\( \text{sec} \)), and cotangent (\( \text{cot} \)).

For the example of \(4900^{\circ}\) or \(220^{\circ}\) in Quadrant III, the sine and cosine functions are negative, while the tangent function remains positive. This means:

• \( \text{sin}(220^{\circ}) = - \text{sin}(40^{\circ}) \)


• \( \text{cos}(220^{\circ}) = - \text{cos}(40^{\circ}) \)


• \( \text{tan}(220^{\circ}) = \text{tan}(40^{\circ})\)

Refer to appropriate trigonometric tables or calculators to find the specific values, then assign the correct signs based on the quadrant. This approach helps accurately determine the trigonometric functions for any angle, regardless of its size.